I did a little study on Lambert w function. Lambert w function is actually a complex function and can also be expressed in form of laurant series. And it is multivalued i.e there are multiple solutions in complex plane but only 3 solutions in real line. About log i am not exactly sure but to say the least i remember this equation from Isc 2nd year, logx=x+x^2/1!+x^3/2!+......... so basically value of log can be found with good degree of precision. And you also can take log for complex numbers. But log being 200year old tradition, I dont really think it has too many flaws.